Substitution Dynamical Systems-Spectral Analysis

Queffélec, Martine

doi:10.1007/bfb0081890

Cited by 588 publications

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“…The most spectacular result was given in the case of a discretized Laplacian with a potential taking on finitely many values (called letters): in such a case the set of gap labels is the Z-module generated by the occurrence probability of all possible finite words found in the sequence defined by the potential. If this sequence is given by a substitution, these occurrence probabilities can be computed explicitely in terms of the incidence matrix of the substitution and of the associated substitution induced on the set of words with two letters [58,9]. The key property in proving such results was the use of the Pimsner-Voiculescu exact sequence [55].…”

Section: Historic Backgroundmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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Section: Historic Backgroundmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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“…In order to answer this question, we recall now a useful construction which can be found in [21]. Let [12] (Theorem 30).…”

Section: The S( (N) Functionsmentioning

confidence: 99%

“…With any symbolic sequence u defined over a finite set, one can also associate the dynamical system (O(u), T), where 0(u) is [18]), that the order of magnitude of the functions ON (u) [21] (ii) ~82 = 1, ~x2 = 0, 02 is a root of unity and Aa,u = 0.…”

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confidence: 99%

Symbolic discrepancy and self-similar dynamics

Adamczewski¹

2004

Annales De L’institut Fourier

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“…In this paper, the two-symbol substitutional sequence used is the RS sequence. RS sequences are two-symbol aperiodic sequences with random-like character [20][21][22]. In its basic form, an RS sequence is first generated from the two-symbol alphabet {−1, 1} via the simple recursive rule…”

Section: Background and Problem Formulationsmentioning

confidence: 99%

Suppression of Backscattering From 2-D Aperiodically-Ordered Thinned Patch Array Using Rudin-Shapiro Sequences

Sallam¹,

Attiya²

2018

PIER M

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Abstract-The discovery of "quasi-crystals," whose X-ray diffraction patterns reveal certain unusual features which do not conform with spatial periodicity, has motivated studies of the wave-dynamical implications of "aperiodic order." This paper discusses various aperiodic configurations generated by Rudin-Shapiro (RS) sequences. These RS sequences constitute ones of the simplest conceivable examples of deterministic aperiodic geometries featuring random-like (dis)order. The scattering properties of aperiodically-ordered thinned 2-D patch arrays based on RS sequences are analyzed by using physical optics approximation. Compared to a periodic case, RS-based antenna array is found to have a substantial reduction in the magnitude of the backscattering component of the scattered signal with half of the elements and the same magnitude of specular reflection. This property is verified by illustrative numerical parametric studies.

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Substitution Dynamical Systems-Spectral Analysis

Cited by 588 publications

References 0 publications

A spectral sequence for theK-theory of tiling spaces

A spectral sequence for theK-theory of tiling spaces

Symbolic discrepancy and self-similar dynamics

Suppression of Backscattering From 2-D Aperiodically-Ordered Thinned Patch Array Using Rudin-Shapiro Sequences

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